I have no idea whether this will help but my '66 (Corvette 327 powered) had the same problem, as apparently, did a lot of GM's of that era. (Does this apply to yours? I have no idea.)
Apparently, the spring inside the solenoid was a bit strong and would get a bit stronger when hot and lead to the clicking/starter not engaging, yadda, yadda, yadda. (I went through the same things you've done and feel your pain.) According to Jon Meyer, GM actually came out with a factory tech sheet regarding a weaker replacement spring as a fix. In searching old AOAI tech articles, Glenn Bell had addressed this problem in issue 94 on page 23. Summit Racing has a kit, p/n SUM-G1750 which is an external relay and wiring kit which basically ensures that the solenoid gets engaged. I put it on about 5 years ago (very simple procedure) and have not had a single hiccup since. Hope this helps.